Investigation of rack cutter geometry and trochoid curves at transverse section in gear manufacturing simulation

The root fillet profile determines the bending strength of a gear tooth. When manufacturing gears by generating- type cutters, the tool tip center of curvature follows a trochoidal path that determines the root profile of the gear. Rack tool tip at transverse section is a part of an ellipse for helical and beveloid gears. This ellipse can be vertical, horizontal or rotated due to existence of helix and cone angles. The envelope of the family of ellipses whose centers are on the trochoid path forms the secondary trochoid curve that determines generated gear actual root fillet profile. None of the studies in the literature include the parametric equations of the secondary trochoid curve for helical and beveloid gear types. Based on the parametric equations of rotated ellipses, this study proposes an approach to obtain the equations of the family of ellipses and secondary trochoids for helical and beveloid gears. Derivatives of primary trochoid curve, rolling angle of gear blank and rotation angle of tip ellipse are used to obtain the corresponding parameter that gives the secondary trochoid point on the enveloping ellipse. Numerical examples of spur, helical, straight beveloid and helical beveloid gears are given to verify and to validate the proposed approach. Results indicate that the proposed approach is a practical way to calculate the secondary trochoid points on the enveloping ellipses.